Idea

an adjunction with UUright adjoint, under certain conditions it is possible to transfer the model structure from CC to a model structure on DD by declaring the fibrations and weak equivalences in DD to be precisely those morphisms whose image under UU are fibrations or weak equivalences, respectively, in CC.

Typically this arises in situations where DD consist of the “same” objects as CC but equipped with extra stuff, structure, property, and UU is the corresponding forgetful functor sending objects in DD to their underlying objects in CC. Then FF is the corresponding free functor.

(meaning: a factorization of the diagonalΔ:A→A×A\Delta : A \to A \times A as a weak equivalence followed by a fibration (under UU) Δ:A→≃P(A)→fibA×A\Delta : A \stackrel{\simeq}{\to} P(A) \stackrel{fib}{\to} A \times A, functorial in AA).

If these conditions are met, then for II (resp. JJ) the set of generating (acyclic) cofibrations in CC, the image set F(I)F(I) (resp. F(J)F(J)) forms the set of generating (acyclic) cofibrations in DD.

Proof

be a pullback diagram in DD, with the bottom morphism a weak equivalence and the right morphism a fibration. We need to show that then also the top morphism ff is a weak equivalence. By definition of transfer, this is equivalent to U(f)U(f) being a weak equivalence in CC.

is a pullback diagram in CC. Since by definition of the transferred model strucure this is still the pullback of a weak equivalence along a fibration, and since CC is assumed to be right proper, it follows that U(f)U(f) is a weak equivalence in CC, hence that ff is a weak equivalence in DD.

Observation

satisfies the conditions of the above proposition so that the model structure on CC is transferred to DD. Consider the case that CC is moreover an SS-enriched model category and that DD can be equipped with the structure of a SS-enriched category that is also SS-powered and copowered.

Assume now that the SS-powering of DD is taken by UU to the SS-powering of CC, in that U(d(s1→s2))=U(d)(s1→s2)U(d^{(s_1 \to s_2)}) = U(d)^{(s_1 \to s_2)}.

Then the transferred model structure and the SS-enrichment on DD are compatible and make DD an SS-enriched model category.

Proof

By the axioms of enriched model category one sufficient condition to be checked is that for s→ts \to t any cofibration in SS and for X→YX \to Y any fibration in DD, we have that the induced morphism

Xt→Xs×YsYt
X^t \to X^s \times_{Y^s} Y^{t}

is a fibration, which is a weak equivalence if at least one of the two input morphisms is. By the induced model structure, this is checked by applying UU. But by assumption UU commutes with the powering, and since UU is a right adjoint it commutes with taking the pullback, so that under UU the morphism is

U(X)t→U(X)s×U(Y)sU(Y)t
U(X)^t \to U(X)^s \times_{U(Y)^s} U(Y)^{t}

which is the morphism induced from U(X)→U(Y)U(X) \to U(Y). That this is indeed an (acyclic) fibration follows now from the fact that CC is an SS-enriched model category.